Objects origin is not moving in edit mode
Pivot Point and Object Origin are different things.  The Object Origin isn't meant to move in edit mode.  The Pivot point depends on the Transform Pivot Point setting.  Take a look at the top bar in the 3D window: 
The second drop down sets the pivot point: 
You haven't screwed up any settings.  You just have the Pivot Point set to &quot;median point&quot;. Because you have selected a cube, the median point happens to be the center of the cube.
If you want to pivot along the object origin, change the drop down to &quot;Individual origins&quot;.
To move the origin, in object mode, Right Click and select &quot;Set Origin&quot; from the popup menu